Theorem rmoeqd 2672

Description: Equality deduction for restricted at-most-one quantifier. (Contributed by Alexander van der Vekens, 17-Jun-2017.)

Hypothesis

Ref	Expression
raleqd.1	|- ( A = B -> ( ph <-> ps ) )

Assertion

Ref	Expression
rmoeqd	|- ( A = B -> ( E* x e. A ph <-> E* x e. B ps ) )

Distinct variable groups:   x, A   x, B

Allowed substitution hints:   ph( x)   ps( x)

Proof of Theorem rmoeqd

Step	Hyp	Ref	Expression
1		rmoeq1 2664	. 2 |- ( A = B -> ( E* x e. A ph <-> E* x e. B ph ) )
2		raleqd.1	. . 3 |- ( A = B -> ( ph <-> ps ) )
3	2	rmobidv 2654	. 2 |- ( A = B -> ( E* x e. B ph <-> E* x e. B ps ) )
4	1, 3	bitrd 187	1 |- ( A = B -> ( E* x e. A ph <-> E* x e. B ps ) )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1343   E*wrmo 2447

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-cleq 2158  df-clel 2161  df-nfc 2297  df-rmo 2452

This theorem is referenced by: (None)